One-dimensional stable probability density functions for rational index 0 < α ≤ 2

Annals of Physics

Volumn 323, Issue 12, 2008, Pages 3000-3019

  Hatzinikitas, Agapitos ^a   Pachos, Jiannis K ^b  

^a UNIVERSITY OF THE AEGEAN   (Greece)

^b UNIVERSITY OF LEEDS   (United Kingdom)

Author keywords

Integral equations; L vy flights; Probability distributions; Special functions

Indexed keywords

EID: 54849428320     PISSN: 00034916     EISSN: 1096035X     Source Type: Journal    
DOI: 10.1016/j.aop.2008.06.004     Document Type: Article

References (30)

1. Lévy P. Calcul des Probabilités (1925), Gauthier-Villars
2. Gnedenko B.V., and Kolmogorov A.N. Limit Distributions for Sums of Independent Random Variables (1968), Addison-Wesley
3. Bardou F., Bouchaud J.P., Aspect A., and Cohen-Tannoudji C. Lévy Statistics and Laser Cooling (2002), Cambridge University Press, London
4. Shlesinger M.F., West B.J., and Klafter J. Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58 (1987) 1100
5. In: Shlesinger M.F., Zaslavsky G.M., and Frisch U. (Eds). Lévy Flights and Related Topics in Physics (1995), Springer-Verlag
6. Nikias C.L., and Shao M. Signal Processing with Alpha-Stable Distributions and Applications (1995), John Wiley and Sons, New York
7. Levandowsky M., White B.S., and Schuster F.L. Random movements of soil amebas. Acta Protozool. 36 (1997) 237
8. Koren T., Lomholt M.A., Chechkin A.V., Klafter J., and Metzler J. Leapover lengths and first passage time statistics. Phys. Rev. Lett. 99 (1999) 160602
9. Mantegna R.N., and Stanley H.E. Econophysics: scaling and its breakdown. J. Stat. Phys. 89 (1997) 469
10. Mainardi F., Luchko Yu., and Pagnini G. The Fundamental Solution of the Space-Time Fractional Diffusion Equation. Fract. Calculus Appl. Anal. 4 2 (2001) 153
11. Mainardi F., Pagnini G., and Saxena R.K. Fox H functions in fractional diffusion. J. Comput. Appl. Math. 178 1-2 (2005) 321
12. Feller W. An Introduction to Probability Theory and its Application. 2nd ed. vol. II (1971), John Wiley and Sons, New York
13. Sato K.-I. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics vol. 68 (2004), Cambridge University Press, United Kingdom
14. Applebaum D. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics vol. 93 (2005), Cambridge University Press, United Kingdom
15. Hughes B.D. Random Walks and Random Environments. Random Walks vol. 1 (1995), Oxford University, United Kingdom
16. X. Ma, C.L. Nikias, On Blind Channel Identification for Impulsive Signal Environments, in: Proc. ICASSP'95, Detroit, MI.
17. V.M. Zolotarev, One-Dimensional Stable Distributions. Am. Math. Soc., Providence, RI, 1986.
18. Fox C. The G and H functions as symmetrical Fourier kernels. Trans. Am. Math. Soc. A98 (1961) 395
19. Mathai A.M., and Saxena R.K. Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Lecture Notes in Mathematics vol. 348 (1970), Springer-Verlag
20. Schneider W.R. Stochastic Processes-Mathematics and Physics II. Lecture Notes in Mathematics vol. 1250 (1985), Springer-Verlag
21. Schneider W.R., and Wyss W. Fractional diffusion and wave equations. J. Math. Phys. 30 (1989) 134
22. West B.J., Grigolini P., Metzler R., and Nonnenmacher T.F. Fractional diffusion and Lévy stable processes. Phys. Rev. E55 (1997) 99
23. Jespersen S., Metzler R., and Fogedby H.C. Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions. Phys. Rev. E59 (1999) 2736
24. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1
25. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives-Theory and Applications (1993), Gordon and Breach, New York
26. Miller K.S., and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons, New York
27. Gradshteyn I.S., and Ryzhik I.M. Table of Integrals, Series, and Products (1994), Academic Press
28. Holtsmark J. Über die Verbreiterung von Spektrallinien. Ann. Phys. 363 (1919) 577
29. Chandrasekhar S. Stochastic problems in physics and astronomy. Rev. Mod. Phys. A15 (1943) 1
30. Luke Y.L. The Special Functions and Their Approximations vol. 1 (1969), Academic Press